scholarly journals On the representations of Leavitt path algebras

2011 ◽  
Vol 333 (1) ◽  
pp. 258-272 ◽  
Author(s):  
Daniel Gonçalves ◽  
Danilo Royer
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals FP-injective semirings, semigroup rings and Leavitt path algebras

2016 ◽  
Vol 45 (5) ◽  
pp. 1893-1906 ◽  
Author(s):  
Marianne Johnson ◽  
Tran Giang Nam
Keyword(s):  
Semigroup Rings ◽  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

2021 ◽  
Vol 588 ◽  
pp. 200-249
Author(s):  
Simon W. Rigby ◽  
Thibaud van den Hove
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Grzegorz Bajor ◽  
Leon van Wyk ◽  
Michał Ziembowski
Keyword(s):  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
Unital Ring ◽  
Commutative Subalgebra ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Commutative Subalgebras

Abstract Considering prime Leavitt path algebras L K ⁢ ( E ) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ ( E ) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ ( E ) {\mathcal{M}_{K}(E)} of L K ⁢ ( E ) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ ( E ) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ ( E ) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring.

scholarly journals Group gradations on Leavitt path algebras

2019 ◽  
Vol 19 (09) ◽  
pp. 2050165 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Directed Graph ◽  
Path Algebra ◽  
Graded Rings ◽  
Leavitt Path Algebra ◽  
Arbitrary Group ◽  
Graded Ring ◽  
Identity Component ◽  
Path Algebras ◽  
Leavitt Path Algebras

Given a directed graph [Formula: see text] and an associative unital ring [Formula: see text] one may define the Leavitt path algebra with coefficients in [Formula: see text], denoted by [Formula: see text]. For an arbitrary group [Formula: see text], [Formula: see text] can be viewed as a [Formula: see text]-graded ring. In this paper, we show that [Formula: see text] is always nearly epsilon-strongly [Formula: see text]-graded. We also show that if [Formula: see text] is finite, then [Formula: see text] is epsilon-strongly [Formula: see text]-graded. We present a new proof of Hazrat’s characterization of strongly [Formula: see text]-graded Leavitt path algebras, when [Formula: see text] is finite. Moreover, if [Formula: see text] is row-finite and has no source, then we show that [Formula: see text] is strongly [Formula: see text]-graded if and only if [Formula: see text] has no sink. We also use a result concerning Frobenius epsilon-strongly [Formula: see text]-graded rings, where [Formula: see text] is finite, to obtain criteria which ensure that [Formula: see text] is Frobenius over its identity component.

scholarly journals Representations of Leavitt path algebras

2020 ◽  
Vol 224 (3) ◽  
pp. 1297-1319
Author(s):  
Ayten Koç ◽  
Murad Özaydın
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras

2018 ◽  
Vol 371 (8) ◽  
pp. 5461-5486 ◽  
Author(s):  
Lisa Orloff Clark ◽  
Cain Edie-Michell ◽  
Astrid an Huef ◽  
Aidan Sims
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals Leavitt path algebras satisfying a polynomial identity

2016 ◽  
Vol 15 (05) ◽  
pp. 1650084 ◽  
Author(s):  
Jason P. Bell ◽  
T. H. Lenagan ◽  
Kulumani M. Rangaswamy
Keyword(s):  
Polynomial Identity ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
Graph Theoretic ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Dimension One ◽  
Kirillov Dimension

Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

scholarly journals Purely infinite simple Kumjian–Pask algebras

2018 ◽  
Vol 30 (1) ◽  
pp. 253-268 ◽  
Author(s):  
Hossein Larki
Keyword(s):  
Commutative Ring ◽  
Path Algebras ◽  
Higher Rank ◽  
Leavitt Path Algebras ◽  
Unital Simple

Abstract Given any finitely aligned higher-rank graph Λ and any unital commutative ring R, the Kumjian–Pask algebra {\mathrm{KP}_{R}(\Lambda)} is known as the higher-rank generalization of Leavitt path algebras. After the characterization of simple Kumjian–Pask algebras by Clark and Pangalela among others, in this article we focus on the purely infinite simple ones. Briefly, we show that if {\mathrm{KP}_{R}(\Lambda)} is simple and every vertex of Λ is reached from a generalized cycle with an entrance, then {\mathrm{KP}_{R}(\Lambda)} is purely infinite. We also prove a dichotomy for simple Kumjian–Pask algebras: If each vertex of Λ is reached only from finitely many vertices and {\mathrm{KP}_{R}(\Lambda)} is simple, then {\mathrm{KP}_{R}(\Lambda)} is either purely infinite or locally matritial. This result covers all unital simple Kumjian–Pask algebras.

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